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Home , Irrational number

Is i irrational? by Colin Foster

I once saw a Key Stage 3 lesson in which pupils were Quite often, statements are not returned to and simplifying algebraic expressions, and where each reformulated as new things are learned. Assumptions question part (a, b, c, …) involved only the letter of that build up that may be helpful for a while but become question, so part b might have been something like problematic later (Tall, 2013). For example, typically 5b – 2b + b – 3b. When some of the pupils reached part i and were working on a question like 8i + 4i – 9i + 3i, make an appearance. An irrational is the idea of ‘irrational’ gets defined long before complexp the teacher stopped the class and suggested to the pupils any number which can’t be written in the form q , where (tongue in cheek, of course) that in that question part p and q are . Perhaps we should have said ‘any they were actually doing something much more advanced. …’? But then what would that have added for They could congratulate themselves that they were doing someone who hasn’t yet met any number that isn’t real, complex numbers, since in maths i actually means an or doesn’t even know that such a number could exist? It’s (Note 1)! the sort of thing the teacher might slip in for the sake of their own mathematical conscience, but which is either Of course, the teacher was not being totally serious not noticed by the pupils or leads to questions which are about this, and had chosen to take an opportunity to introduce the pupils to the problem of -rooting lesson. (If you are introducing irrational numbers today, a , saying that we use the letter i as youlikely might to divert not want significantly today also from to be the the main lesson point where of youthe a ‘made-up number’ for − . But it got me thinking 1 have to introduce the idea of an imaginary number.) So whether in any sense it could be true to say that they one day, years later, when a pupil asks, “Is i irrational?” were ‘doing complex numbers’? On the face of it, their there may be a bit of work to do in sorting it out. activity didn’t look much different from that of a further It may be that this question arises partly because i is the numbers, simplifying expressions like 3 + 8i – 5 – 2i. Ismathematics i + i = 2i only student adding in theirimaginary first lessonnumbers on if complex I think also be provoked by the cartoon (Fig. 1) that occasionally first letter of irrational (as well as imaginary). It might about the fact that i = −1 as I do it? Of course, it is only a convention to use i for −1; any other part of that the numbers i and , and has i saying to , “Be rational!” question could equally well have been thought of as anddoes the saying rounds to i “Getamong real!” sixth This formers, joke depends which personifies on the fact involving complex numbers, as of course nowhere did that is real but notπ rational, but mightπ be understood the book specify that the quantities were all real! as suggestingπ that i is rational but not real, otherwise i is beingπ hypocritical! So is i rational or irrational? Sometimes we do not always state clearly what we mean. For example, we might say that although we can factorize x2 – 1, as (x + 1)(x – 1), we can’t factorize x2 + 1, for example. However, just changing the letter name from x to z might suggest otherwise. On seeing z2 + 1, we might think in terms of working over the complex numbers, rather than over the reals, and so say that we could factorize it, as (z + i) (z – i) (Note 2). So the answer to “Is x2 + 1 irreducible?” depends on the of numbers which we consider the

choice of a letter x could push us towards considering onlycoefficients real numbers. to belong to, and something as slight as a Fig. 1 A number joke

Mathematics in School, January 2018 The MA website www.m-a.org.uk 31

31-33-Foster QA is i irrational.indd 31 22/12/2017 09:14 … –3 < –2 < –1 < 0 < 1 < 2 < 3 … be expressed as the ratio of two integers. The only square Similarly, the imaginary axis is also a line, so presumably rootsBy our of definition integers whichabove, are i is rational not rational, are the since square it cannot roots we can write: of positive square numbers (and zero). Pupils might think that although –5 is not rational, –9 might possibly be, … –3i < –2i < –i < 0 < i < 2i < 3i … since 9 is a square number. However, writing = p , where –9 q But in general, the non-reals have two parts to them – q p2 = –9q2, which has no solutions for real p they are like ‘2-dimensional numbers’, so they can’t be and q, except p = q = 0, which contradicts the statement that put into order along a 1-dimensional line. q ≠ 0, leads to However, this is not right at all, as I only realized much So i is not rational — so is it irrational then? Not so fast! ≠ 0. So based on this we cannot call i rational. later on! The symmetry between the reals and the purely ‘Irrational’ is normally taken to refer to a real number imaginaries is deceptive. Maybe the use of vertical number which is not rational, so on this basis i is neither rational lines in school (e.g. temperature scales for working with nor irrational. Just as it doesn’t make sense to ask whether negative numbers) contributes to this easy false idea? But 2 is odd or even, because it is neither of the form 2n nor 3 the Argand diagram is not merely a horizontal number line of the form 2n + 1 (where n is an ), it doesn’t make coupled with a vertical number line. The problem is that sense to ask whether i is rational or irrational. (Pupils we cannot say that 3i > 2i. This is quite counterintuitive: may think of ‘odd’ as any number that isn’t even but surely, whatever i is, 3 lots of it is more than 2 lots of it? But not state carefully that ‘number’ in this context means no! Let’s suppose that 3i > 2i. If we subtract 2i from both ‘integer’.) Rational and irrational are opposites, so it may sides, we get i > 0. But that cannot be true. If we multiply seem that a number which isn’t one must be the other, both sides of this inequality by i, since we are supposing and this is true across the reals, just as all integers are that i is positive (i > 0), the inequality sign will stay the either even or odd. However, the symmetry between same way round, giving i2 evens and odds is really quite unlike that between the know that i2 , giving us –1 > 0, which is clearly rationals and irrationals. We can specify what all the false. This means that we > must0. But haveby the been definition wrong of to i rationals are like ( with integer numerator and supposewe that i > 0,= or –1 that 3i > 2i. non-zero integer denominator), but the irrationals (of which, in a sense, there are far more) include not just the So we might think that, as it came out the wrong way surds but all kinds of strange things like e and , and even round at the end, with –1 > 0 instead of –1 < 0, we must lots of numbers which we can’t express in closed form. just have started the wrong way round at the beginning. The irrationals feel like the contents of a bin πinto which We have found that i isn’t positive, so it must be negative? we put all the badly-behaved numbers – and for this Let’s try that instead. Starting with i < 0, if we multiply reason it might feel like i really ought to go in that bin! both sides of this inequality by i, then since i is supposed to be negative now, this will reverse the inequality sign. Another problematic issue that arises with imaginary That means that we get i2 > 0 again, just like before, which numbers is to do with ordering. We know that for any we have just shown can’t be true! two real numbers, a and b, exactly one of the statements If i > 0 is false, and i < 0 is also false, perhaps we should a = b, a > b, a < b try i = 0? This seems unlikely. Starting with i = 0, we can must be true (the law of trichotomy). However, we can’t multiply both sides of this by i, and we get i2 = 0, do this with complex numbers. They can’t be ordered in which means –1 = 0, which is again false, of course. So we a useful way. If we take, for instance, 2 + 3i and 3 + 2i, we must have been wrong to say i = 0. can’t say This means that i is not greater than zero, or less than zero, 2 + 3i = 3 + 2i or equal to zero. We can’t use inequality signs between or 2 + 3i > 3 + 2i non-real numbers. I think I didn’t realize that this applied or 2 + 3i < 3 + 2i. to the purely imaginary numbers until at university I noticed that writing “Let > 0” also meant by implication None of those three statements is true. If we think of that was real. But I should have understood this at school. the modulus of the two complex numbers, then we can This raises the question εof what the imaginary axis on say |2 + 3i| = |3 + 2i|, since the modulus of a complex an Argandε diagram is doing? Isn’t the imaginary axis an number is a real number, and we know that we are OK ordering of the purely imaginary numbers? If they really ordering real numbers. can’t be ordered, would it make just as much sense if, When at school, I mistakenly thought that this was instead of numbering the imaginary axis 0, i, 2i, 3i, 4i, etc. obvious. The complex plane is 2-dimensional, I thought, (Fig. 2), we numbered it haphazardly, going up so how can you possibly put all those points in order, 2 −−7i, i, 5i, 4976i, etc? whereas the real axis is a line, with the points neatly 3 ordered along it. Just thinking of the integers, we have:

32 Mathematics in School, January 2018 The MA website www.m-a.org.uk

31-33-Foster QA is i irrational.indd 32 22/12/2017 09:14 5i and-replace in all of mathematics and swapping all the 1s for –1s would create a huge mess! Again, I remember 4i thinking at school that the pure imaginaries were just like 3i the reals; that there was a symmetry between the two. I remember thinking that if we had ‘discovered imaginary 2i

i children would count their sweets as i, 2i, 3i, … (although nonumbers doubt lazily first’ dropping we might the have i’s!). called I think them the apparent the reals two- and 0 -5 -4 -3 -2 -1 012345 way symmetry of the Argand diagram contributed to this

-i misconception on my part. But the left–right symmetry of the Argand diagram is quite unlike the top–bottom -2i symmetry, so I really was wrong about quite a lot of things!

-3i 5

-4i 4

-5i 3

Fig. 2 2 in an Argand diagram A common way of labelling the axes 1 Eventually I realized that what we are doing with the 0 imaginary axis on an Argand diagram is plotting the -5 -4 -3 -2 -1 0 12345

imaginary parts of the purely imaginary numbers – and -1 of course those imaginary parts are real! So, for a number like 4i, we plot Im(4i) = 4 at (0, 4). So it’s actually real -2

numbers that we are plotting on the Argand plane, and -3 real numbers can be ordered. The fact that zero lies on the imaginary axis should really have alerted me to this, -4 since 0 is not an imaginary number (even if you write it -5 as 0i)! Fig. 3 Perhaps this means that instead of the labelling as in and imaginary parts Figure 2 (which is common in textbooks), we should be Labelling the axes of an Argand diagram with the real a bit more careful and do it as in Figure 3? The z = x +iy is represented by the point (x, y), and of Notes course here both x and y are real. Perhaps this also means 1. A cynic could argue that this was not ‘actually’ the case, since the that we shouldn’t refer to the ‘positive imaginary axis’ in letter i was italic, and imaginary i is normally printed non-italic! the Argand diagram? Or, if we label the real axis with an 2. This is actually a bit subtle, because the issue is not really whether the arrow on the right-hand end, to indicate direction, maybe variable x or z might be non-real so much as whether the coefficients we shouldn’t do this at the top of the ‘imaginary’ axis? might. Although, I suppose, we could say that what we mean by the ‘positive imaginary axis’ is the part of the imaginary Reference axis relating to imaginary numbers whose imaginary parts are positive! Tall, D. 2013 How Humans Learn to Think Mathematically: Exploring the Three Worlds of Mathematics, Cambridge University Press, All of this is closely related to the symmetry between the Cambridge. two square roots of –1. We shouldn’t call i the ‘positive ’ of –1 and –i the ‘negative square root’ of –1. Colin Foster was recently interviewed on the Mr Barton podcast: see http://www.mrbartonmaths.com/ Some would say that we shouldn’t even write i1=−, because the radical symbol conventionally means ‘the blog/colin-foster-mathematical-etudes-confidence- positive square root of …’. As we have seen, despite and-questioning appearances, neither i nor –i is positive or negative, since Complex numbers; it makes no sense to say that either of them is greater or Keywords: less than zero. In fact, neither i nor –i is more fundamental, Argand diagram; and we could go through the whole of mathematics Imaginary numbers; Inequalities. Author Colin Foster, School of Education, University of Leicester, replacing all the i’s with –i’s and everything would work 21 University Road, Leicester LE1 7RF. not the case with 1 and –1. e-mail: [email protected] website: www.foster77.co.uk For example, 1 × 1 = 1 but –1 × out fine! Note that this is –1 ≠ –1. Doing a search- Mathematics in School, January 2018 The MA website www.m-a.org.uk 33

31-33-Foster QA is i irrational.indd 33 22/12/2017 09:14